# Normalizers of subgroups of subsystems of Lie groups.

Leave $$G$$ be a group of complex semisimple lies, and let $$H$$ be a subgroup corresponding to a subset of the extended Dynkin diagram of $$G$$ (to the Borel – of Siebenthal). I would like to know if there is a recipe to calculate the normalizer of $$H$$. My feeling is that this must be known, but I could not find anything.

To be specific, consider this example. Leave $$G = E_7$$ (simply connected). There is a subgroup of type $$A_7$$, which has an index $$2$$ Inside your normalizer. This corresponds to the involution of the Dynkin type diagram. $$A_7$$, which one can check respects the highest root of $$E_7$$ And so it respects the extended Dynkin type diagram. $$E_7$$. On the other hand, consider the type subgroup $$D_6 times A_1$$. The index of the subgroup in its normalizer is at most $$2$$, because the Dynkin diagram has a group of automorphism $$2$$. It is clear that the obvious involution of the Dynkin diagram does not extend to an involution of the extended Dynkin diagram $$E_7$$, because the highest root is sent to some other root. Does this imply that the subgroup is self-normalizing? I hope that this subgroup does not self-normalize.